Area of a Sector

A sector is a portion of a circle bounded by two radii and the arc joining their extremities. It is thus a form of triangle with a covered base. In the figure the portion AOB is a sector. Arc AB is called the arc of the sector and \angle AOB is called the angle of the sector.


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The area of a sector of a circle is equal to a fraction of the area of the circle determined by dividing the size of the angle by {360^ \circ }. Thus, if the angle of the sector is {90^ \circ }, the area of the sector is \frac{{90}}{{360}} or \frac{1}{4} the area of the circle, that is, i.e., the area of a sector bears that portion to the area of the circle which its angle bears to 4 right angles i.e., {360^ \circ }.

If the angle \angle AOB is given in degrees, say{N^ \circ }, then

            Area of the sector, A = \frac{{\pi {r^2}}}{{360}} \times  {N^ \circ }

            Length of the arc, l = \frac{{2\pi r}}{{360}} \times  {N^ \circ }

If the angle \angle AOB is given in radians, say \theta radians, then

            Area of the sector, A = \frac{{\pi {r^2}}}{{360}} \times  {\left( {\frac{{180}}{\pi } \times \theta } \right)^ \circ } =  \frac{1}{2}{r^2}\theta

If the length l of the arc and the radius r of the circle are given, then

            Area of the sector, A = \frac{1}{2}{r^2} \times  \theta = \frac{1}{2}{r^2} \times  \frac{1}{l}   where \theta =  \frac{l}{r}

                                           A =  \frac{1}{2}r \times l

Example:

Find the area of a sector of {60^ \circ } in a circle of radius 10cm.

Solution:

The sector AOB has angle {60^ \circ }.


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            Its area is equal to \frac{{60}}{{360}} \times \pi {r^2}
            Since radius  = 10cm
            Therefore A = \frac{1}{6} \times 3.1416 \times {(10)^2} =  52.36 Square

Example:

The minute hand of a clock is 12cw long. Find the area which is described on the clock face between 6A.M to 6.20A.M.

Solution:

Given that, Length of minute hand, radius  = 12cm
            \therefore 60Minutes   = {360^ \circ }

            \therefore 20Minutes   = {120^ \circ }

            Since, Area of sector  = \frac{{\pi {r^2}}}{{360}} \times  {N^ \circ }

                                                =  \frac{{3.1416 \times {{(12)}^2}}}{{360}} \times {120^ \circ }

                                                =  \frac{1}{3} \times 3.1416 \times 144 = 150.7 Square cm

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